Multiple choice

The real values of $a$ for which the quadratic equation $2x^2 - (a^3 + 8a - 1) x + a^2 - 4a = 0$ possesses roots of opposite signs are given by :

  1. $a > 6$
  2. $a > 9$
  3. $0 < a <4$
  4. $a < 0$
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C Correct answer
Explanation

For a quadratic equation to have roots of opposite signs, the product of the roots (c/a) must be negative. Here, (a^2 - 4a) / 2 < 0. So a(a-4) < 0. This holds when 0 < a < 4.

AI explanation

For a quadratic equation to have roots of opposite signs, the product of the roots must be negative, meaning the constant term divided by the leading coefficient must be less than zero. For the equation 2x squared minus the quantity a cubed plus 8a minus 1 times x plus a squared minus 4a equals 0, we set the quantity a squared minus 4a divided by 2 to be less than zero. This simplifies to a times the quantity a minus 4 being less than zero, which holds true when a is between 0 and 4.

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